Differential topology in \(4=3+1\)  dimensions is dramatically different from all other dimensions. An object that perfectly encapsulates this radical difference is the homology cobordism group \(\Theta _n\)  consisting of smooth homology \(n\) -spheres up to homology cobordism with connected sum as operation: it was shown by Kervaire [1]} that \(\Theta _n\)  is always a finite group for \(n\ne 3\) , while Furuta [2]} proved that \(\Theta _3\)  contains a subgroup isomorphic to \(\mathbb {Z}^{\infty }\)  (hence in particular it is not finitely generated).
